If two dice are rolled thrice, then the mean of X where X denotes the number of times even numbers obtained on both dice is

Options

A. `(1)/(4)`
B. `(3)/(4)`
C. `(1)/(2)`
D. `(1)/(5)`

Answer

To solve the problem, we need to find the mean of the random variable \( X \), which denotes the number of times even numbers are obtained on both dice when two dice are rolled thrice.

### Step-by-Step Solution:

1. **Identify the Total Outcomes When Rolling Two Dice:**
Each die has 6 faces, so when rolling two dice, the total number of outcomes is:
\[ 6 \times 6 = 36 \]

2. **Identify the Favorable Outcomes for Getting Even Numbers:**
The even numbers on a die are 2, 4, and 6. We need to find the pairs of outcomes where both dice show even numbers. The pairs are:
- (2, 2)
- (2, 4)
- (2, 6)
- (4, 2)
- (4, 4)
- (4, 6)
- (6, 2)
- (6, 4)
- (6, 6)

Counting these pairs, we find there are 9 favorable outcomes.

3. **Calculate the Probability of Getting Even Numbers on Both Dice:**
The probability \( P \) of getting even numbers on both dice in a single roll is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{9}{36} = \frac{1}{4} \]

4. **Determine the Mean of the Random Variable \( X \):**
Since the dice are rolled 3 times, the expected value (mean) of \( X \), which represents the number of times even numbers are obtained on both dice, can be calculated as:
\[ E(X) = n \cdot P = 3 \cdot \frac{1}{4} = \frac{3}{4} \]

5. **Final Answer:**
The mean of \( X \) is \( \frac{3}{4} \).

### Conclusion:
Thus, the mean of \( X \) where \( X \) denotes the number of times even numbers are obtained on both dice when rolled thrice is \( \frac{3}{4} \). The correct option is **B) \( \frac{3}{4} \)**.